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Find the equation of the line perpendicular to \(\displaystyle y = x + 7\) that contains \(\displaystyle \left( -6, \ -1\right)\).


The slope of the line is \(\displaystyle m = 1\) so the perpendicular line has slope \(\displaystyle m=-1\). The equation has the form \(\displaystyle y=-1x+b\). Using the point \(\displaystyle \left( -6, \ -1\right)\) gives the equation \(\displaystyle -1=-1\left(-6\right)+b\) Solving for \(\displaystyle b\) gives \(\displaystyle b = -7\). The equation of the perpendicular line is \(\displaystyle y = - x - 7\).

Download \(\LaTeX\)

\begin{question}Find the equation of the line perpendicular to $y = x + 7$ that contains $\left( -6, \  -1\right)$. 
    \soln{9cm}{The slope of the line is $m = 1$ so the perpendicular line has slope $m=-1$. The equation has the form $y=-1x+b$. Using the point $\left( -6, \  -1\right)$ gives the equation $-1=-1\left(-6\right)+b$ Solving for $b$ gives $b = -7$.  The equation of the perpendicular line is $y = - x - 7$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas
<p> <p>Find the equation of the line perpendicular to  <img class="equation_image" title=" \displaystyle y = x + 7 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20x%20%2B%207%20" alt="LaTeX:  \displaystyle y = x + 7 " data-equation-content=" \displaystyle y = x + 7 " />  that contains  <img class="equation_image" title=" \displaystyle \left( -6, \  -1\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20-6%2C%20%5C%20%20-1%5Cright%29%20" alt="LaTeX:  \displaystyle \left( -6, \  -1\right) " data-equation-content=" \displaystyle \left( -6, \  -1\right) " /> . </p> </p>
HTML for Canvas
<p> <p>The slope of the line is  <img class="equation_image" title=" \displaystyle m = 1 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%201%20" alt="LaTeX:  \displaystyle m = 1 " data-equation-content=" \displaystyle m = 1 " />  so the perpendicular line has slope  <img class="equation_image" title=" \displaystyle m=-1 " src="/equation_images/%20%5Cdisplaystyle%20m%3D-1%20" alt="LaTeX:  \displaystyle m=-1 " data-equation-content=" \displaystyle m=-1 " /> . The equation has the form  <img class="equation_image" title=" \displaystyle y=-1x+b " src="/equation_images/%20%5Cdisplaystyle%20y%3D-1x%2Bb%20" alt="LaTeX:  \displaystyle y=-1x+b " data-equation-content=" \displaystyle y=-1x+b " /> . Using the point  <img class="equation_image" title=" \displaystyle \left( -6, \  -1\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20-6%2C%20%5C%20%20-1%5Cright%29%20" alt="LaTeX:  \displaystyle \left( -6, \  -1\right) " data-equation-content=" \displaystyle \left( -6, \  -1\right) " />  gives the equation  <img class="equation_image" title=" \displaystyle -1=-1\left(-6\right)+b " src="/equation_images/%20%5Cdisplaystyle%20-1%3D-1%5Cleft%28-6%5Cright%29%2Bb%20" alt="LaTeX:  \displaystyle -1=-1\left(-6\right)+b " data-equation-content=" \displaystyle -1=-1\left(-6\right)+b " />  Solving for  <img class="equation_image" title=" \displaystyle b " src="/equation_images/%20%5Cdisplaystyle%20b%20" alt="LaTeX:  \displaystyle b " data-equation-content=" \displaystyle b " />  gives  <img class="equation_image" title=" \displaystyle b = -7 " src="/equation_images/%20%5Cdisplaystyle%20b%20%3D%20-7%20" alt="LaTeX:  \displaystyle b = -7 " data-equation-content=" \displaystyle b = -7 " /> .  The equation of the perpendicular line is  <img class="equation_image" title=" \displaystyle y = - x - 7 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20-%20x%20-%207%20" alt="LaTeX:  \displaystyle y = - x - 7 " data-equation-content=" \displaystyle y = - x - 7 " /> . </p> </p>